The justice system in America is different from ours, and also varies from state to state, so I don't see how you can cite that by way of comparison.
There are substantial similarities, which isn't surprising given that both share the same common law roots. Things like the presumption of innocence are the same. Also - and in direct relation to your statement - forensic science is the same, and evidence is handled in substantially similar ways.
Adjusted wrongful conviction rates aren't so different between the UK and US either. So that makes it perfectly valid to draw comparisons between the two.
A recent study showed that death penalty convictions in the US had an error rate of about 4%. That's shocking. Even if the UK justice system was 2 times better, that'd mean that out of every 100 convictions leading to the death penalty, 2 people would be wrongfully executed.
So why did the murder rate start to increase in the years following its abolition?
Correlation != causation.
Also I consider myself a sane person, but if I was convinced beyond all shadow of doubt that a person was guilty of murder, then I would see the death penalty as being appropriate, and I'm sure applies to a lot of people.
The problem lies in your first statement: shadow of a doubt. How does one codify that?